Question: $f(x) = \begin{cases} 9 & \text{if } x = 1 \\ 4x^{2}-3 & \text{otherwise} \end{cases}$ What is the range of $f(x)$ ?
First consider the behavior for $x \ne 1$ Consider the range of $4x^{2}$ The range of $x^2$ is $\{\, y \mid y \ge 0 \,\}$ Multiplying by $4$ doesn't change the range. To get $4x^{2}-3$ , we subtract $3$ So the range becomes: $\{\, y \mid y ≥ -3 \,\}$ If $x = 1$, then $f(x) = 9$. Since $9 ≥ -3$, the range is still $\{\, y \mid y ≥ -3 \,\}$.